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Regular and Chaotic Dynamics, 2011, Volume 16, Issue 5, Pages 496–503
DOI: https://doi.org/10.1134/S1560354711050066
(Mi rcd465)
 

This article is cited in 4 scientific papers (total in 4 papers)

Three and Four-body Systems in One Dimension: Integrability, Superintegrability and Discrete Symmetries

Claudia Chanua, Luca Degiovannib, Giovanni Rastellib

a Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, via Cozzi 53, 20126 Milano, Italy
b Formerly at Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy
Citations (4)
Abstract: Families of three-body Hamiltonian systems in one dimension have been recently proved to be maximally superintegrable by interpreting them as one-body systems in the three-dimensional Euclidean space, examples are the Calogero, Wolfes and Tramblay Turbiner Winternitz systems. For some of these systems, we show in a new way how the superintegrability is associated with their dihedral symmetry in the three-dimensional space, the order of the dihedral symmetries being associated with the degree of the polynomial in the momenta first integrals. As a generalization, we introduce the analysis of integrability and superintegrability of four-body systems in one dimension by interpreting them as one-body systems with the symmetries of the Platonic polyhedra in the four-dimensional Euclidean space. The paper is intended as a short review of recent results in the sector, emphasizing the relevance of discrete symmetries for the superintegrability of the systems considered.
Keywords: superintegrability, higher-degree first integrals, discrete symmetries, Tremblay-Turbiner–Winterniz system.
Received: 11.11.2010
Accepted: 27.02.2011
Bibliographic databases:
Document Type: Article
Language: English
Citation: Claudia Chanu, Luca Degiovanni, Giovanni Rastelli, “Three and Four-body Systems in One Dimension: Integrability, Superintegrability and Discrete Symmetries”, Regul. Chaotic Dyn., 16:5 (2011), 496–503
Citation in format AMSBIB
\Bibitem{ChaDegRas11}
\by Claudia Chanu, Luca Degiovanni, Giovanni Rastelli
\paper Three and Four-body Systems in One Dimension: Integrability, Superintegrability and Discrete Symmetries
\jour Regul. Chaotic Dyn.
\yr 2011
\vol 16
\issue 5
\pages 496--503
\mathnet{http://mi.mathnet.ru/rcd465}
\crossref{https://doi.org/10.1134/S1560354711050066}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2844860}
\zmath{https://zbmath.org/?q=an:1309.70021}
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  • https://www.mathnet.ru/eng/rcd/v16/i5/p496
  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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